# The golden section

April 15, 2002 5:00 PM Subscribe

The golden section (math, graphics) is an important relation used by artists and mathematicians, among others. I'm curious if any of you have good examples of recent use.

The golden ratio aka the golden mean aka phi (for you Google searchers out there). I had recalled that Mondrian used this ratio in his work.

posted by vacapinta at 5:31 PM on April 15, 2002

posted by vacapinta at 5:31 PM on April 15, 2002

I was just reading a book about long-term messages (which I was inspired to seek out and purchase after reading an earlier discussion) - and one of the interesting things the author suggested was that the golden section was such a beautiful, innate structure in both academic mathematics and organic nature, that the recognition of it could be a good way to 'break the ice' in communication with an advanced alien society - the assumption being that if they were sufficiently advanced, there's no way they could miss the 'golden ratio'.

posted by kokogiak at 5:34 PM on April 15, 2002

posted by kokogiak at 5:34 PM on April 15, 2002

I go through phases of Golden Section obsession (I once formulated the shape of Utah using it). Most of my books are still packed from moving, but it's really all over the place in nature. From the Fibbonacci series to chromatic musical scales to sunflowers to printed page sizes. It can and has been used in so many different ways, (art, architecture, statistics, geometry, music, genetics, anatomy, etc...) was there anything specific you were looking for?

posted by joemaller at 5:39 PM on April 15, 2002

posted by joemaller at 5:39 PM on April 15, 2002

kokogiak: That sounds interesting. What book is that?

posted by vacapinta at 5:51 PM on April 15, 2002

posted by vacapinta at 5:51 PM on April 15, 2002

I am assuming (following the links ) that it is this book. I've been on a non-fiction bender, I think I'll check it out. Thanks again (and again, and...) MeFi.

posted by anathema at 5:59 PM on April 15, 2002

posted by anathema at 5:59 PM on April 15, 2002

Upon closer examination, this book seems to possibly be on the border between fiction and non-fiction. Back to topic...

posted by anathema at 6:05 PM on April 15, 2002

posted by anathema at 6:05 PM on April 15, 2002

That's the first construction I've seen calling one to "Bisect the damn thing".

BTW, the golden mean/section shows up in the limit in the Fibonacci sequence, by taking successive ratios... (like 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ... etc.)

posted by meep at 6:34 PM on April 15, 2002

BTW, the golden mean/section shows up in the limit in the Fibonacci sequence, by taking successive ratios... (like 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ... etc.)

posted by meep at 6:34 PM on April 15, 2002

The absolute best golden section and Fibonacci sequence website is this one from Dr. Ron Knott of the University of Surrey. I kid you not, I have read every single page on this website. It is incredibly interesting stuff.

After reading the Surrey site a while back, I created this Fibonacci spiral using the now outdated VML. Here's a screen shot for those without VML support. Also, last week I designed a computer controlled mill piece of the Fibonacci spiral for my engineering class.

I guess I'm like you, Joe. I go through the obsession every once in a while, too.

posted by pheideaux at 6:35 PM on April 15, 2002

After reading the Surrey site a while back, I created this Fibonacci spiral using the now outdated VML. Here's a screen shot for those without VML support. Also, last week I designed a computer controlled mill piece of the Fibonacci spiral for my engineering class.

I guess I'm like you, Joe. I go through the obsession every once in a while, too.

posted by pheideaux at 6:35 PM on April 15, 2002

Howard Suber, a film historian and prof at UCLA has posited that the Golden Section is used for a great deal of storytelling, particularly in screenwriting. There is a point, which any screenwriting how to book will mention, 2 thirds of the way thru the script, where the action takes a twist: the protaganist who previously "reacted" will begin to "act". Suber calls this the one hour pivot point as it usually falls 60 minutes into a 90 minute movie (ie, at the golden section).

Suber mentions this on what I believe is the greatest film commentary ever recorded: Criterion's laserdisc for The Graduate. Though out of print, occasionally copies can be found on ebay. I highly recommend it. I learned more about storytelling and filmmaking from Suber's 105 minute commentary than I did in four years of film school.

posted by dobbs at 6:47 PM on April 15, 2002

Suber mentions this on what I believe is the greatest film commentary ever recorded: Criterion's laserdisc for The Graduate. Though out of print, occasionally copies can be found on ebay. I highly recommend it. I learned more about storytelling and filmmaking from Suber's 105 minute commentary than I did in four years of film school.

posted by dobbs at 6:47 PM on April 15, 2002

It can be found in our field of view, I believe. I use it all the time I need to define some arbitrary non equal proportion of things, as it tends to make nice balanced compositions and parts. When you get to subdividing subdivisions enough though, the golden relationship of the part to the whole becomes insignificant. We don't like it

The reason that so many natural objects display this phenomena makes sense when you consider that they grow along a fibonacci sequence, and two consecutive fibonacci numbers of course tends to (1+root five)/2 as is apparent when you divide, say 1 by two, and then 8 by 5, and then 13 by 8, as meep said.

posted by Settle at 7:03 PM on April 15, 2002

*that*much. I've always been more interested in the golden triangle, which is the triangle formed by connecting a vertex of a pentagon with the endpoints of the segment opposite, resulting in rational angle measures. You can actually subdivide this triangle in the same way you subdivide the golden rectangle, and they form the same spiral.The reason that so many natural objects display this phenomena makes sense when you consider that they grow along a fibonacci sequence, and two consecutive fibonacci numbers of course tends to (1+root five)/2 as is apparent when you divide, say 1 by two, and then 8 by 5, and then 13 by 8, as meep said.

posted by Settle at 7:03 PM on April 15, 2002

I know the Charles and Ray Eames were really interested in classical proportions and the golden mean. I'm looking on the web for something about their use of it...

posted by evanizer at 7:20 PM on April 15, 2002

posted by evanizer at 7:20 PM on April 15, 2002

Random trivial item: your credit card is a golden rectangle. It supposedly is just one more thing to make it more appealing to use. Da Vinci was supposed to have used it alot too.

posted by skechada at 7:46 PM on April 15, 2002

posted by skechada at 7:46 PM on April 15, 2002

Two excellent books relating to the golden section:

The Power of Limits

The Geometry of Art and Life

The first one I found a year or two ago, in it the author maps everything from vases to fish along golden rectangle proportions.

I carried a xeroxed chapter from Geometry of Art and Life through several years of college before buying a copy. It's got much more math and worse typography.

I kinda wanted to link myself as an Amazon referrer but resisted the temptation

posted by joemaller at 8:59 PM on April 15, 2002

The Power of Limits

The Geometry of Art and Life

The first one I found a year or two ago, in it the author maps everything from vases to fish along golden rectangle proportions.

I carried a xeroxed chapter from Geometry of Art and Life through several years of college before buying a copy. It's got much more math and worse typography.

I kinda wanted to link myself as an Amazon referrer but resisted the temptation

posted by joemaller at 8:59 PM on April 15, 2002

There is a woman who makes baskets using the Fibonacci numbers as a basis for the pattern and shape. I saw it on HGTV, and she's apparently got a website, too. They're really cool, if a bit expensive.

posted by eilatan at 9:03 PM on April 15, 2002

posted by eilatan at 9:03 PM on April 15, 2002

woolgathering: is the golden rectangle roughly or exactly equal to our (humans) field of vision? Wouldn't that explain our apparently hard-wired preference for the proportion?

posted by yhbc at 9:08 PM on April 15, 2002

posted by yhbc at 9:08 PM on April 15, 2002

reviewing the thread again (I hate when I do that) - is what I thought was my sudden insight what you were referring to above, Settle?

posted by yhbc at 9:12 PM on April 15, 2002

posted by yhbc at 9:12 PM on April 15, 2002

Settle scares me.

posted by obiwanwasabi at 9:40 PM on April 15, 2002

posted by obiwanwasabi at 9:40 PM on April 15, 2002

Yes. Although only about 20% of your field of view is really what you look at, and of course the total shape of your field of view changes by way of convergence and divergence. Field of view is an angular measure of course, and it is further affected by imperfections in your own eyes.

The reason, however, this is a hardwired preference is probably more related to the fact that this is one of the simplest recursively definable "sections" you can take of anything, and the most likely for generative biological systems to end up with. To take half of something that thing must already be complete, but the golden section arises naturally when things are growing out. I seem to remember euler's constant crops up sometimes too.

posted by Settle at 9:42 PM on April 15, 2002

The reason, however, this is a hardwired preference is probably more related to the fact that this is one of the simplest recursively definable "sections" you can take of anything, and the most likely for generative biological systems to end up with. To take half of something that thing must already be complete, but the golden section arises naturally when things are growing out. I seem to remember euler's constant crops up sometimes too.

posted by Settle at 9:42 PM on April 15, 2002

Now I'm scared. Say something dopey and juvenile, so I know it's you.

posted by yhbc at 9:44 PM on April 15, 2002

posted by yhbc at 9:44 PM on April 15, 2002

Settle is a RISD student. He just finished up the drunken haze known as Wintersession and now has to commit his mind to things that are substantive rather merely pooting out art and propping it up with words like juxtapose and symbiotic. This explains his mood swing. As an alum, it all makes sense to me.

posted by machaus at 9:51 PM on April 15, 2002

posted by machaus at 9:51 PM on April 15, 2002

Thank you all for the links.

Although, it almost deserves its own thread, I cant help but mention The Encyclopedia of integer sequences

It's useful for comprehending numbers that seemed to have materialized in your dreams. Or, type some random numbers and discover just how un-random they are.

posted by vacapinta at 10:00 PM on April 15, 2002

Although, it almost deserves its own thread, I cant help but mention The Encyclopedia of integer sequences

It's useful for comprehending numbers that seemed to have materialized in your dreams. Or, type some random numbers and discover just how un-random they are.

posted by vacapinta at 10:00 PM on April 15, 2002

* remembers seeing the RISD campus, and all the happy young people *

* suffers evil flashback to time spent lecturing waitresses on the intricacies of the Uniform Commercial Code while studying for the bar exam *

* understands, but wonders about career choice (again) *

* says the hell with it, goes to bed confident in the future of society *

posted by yhbc at 10:04 PM on April 15, 2002

* suffers evil flashback to time spent lecturing waitresses on the intricacies of the Uniform Commercial Code while studying for the bar exam *

* understands, but wonders about career choice (again) *

* says the hell with it, goes to bed confident in the future of society *

posted by yhbc at 10:04 PM on April 15, 2002

Ha ha ha ha substantiative, what like mefi? I have classes at 9:40 tommorow and I get out early. This school is easy. For want of stuff to do I've been working on my ink wash technique. A good ink-wash technique is one of my turn-ons, and I think anyone who knows what I'm talking about agrees.

also:

www.hotendotey.com

www.sanscomic.com

There, dopey and juvenile. And I'm on amphetamines, I don't drink. That stuff's poison.

posted by Settle at 10:34 PM on April 15, 2002

also:

www.hotendotey.com

www.sanscomic.com

There, dopey and juvenile. And I'm on amphetamines, I don't drink. That stuff's poison.

posted by Settle at 10:34 PM on April 15, 2002

and you're damn right that deserves its own link, vacapinta.

posted by Settle at 10:37 PM on April 15, 2002

posted by Settle at 10:37 PM on April 15, 2002

*and you're damn right that deserves its own link, vacapinta*

Ok, you're on. Liquor is quicker, Settle.

posted by vacapinta at 11:15 PM on April 15, 2002

Chiming in again late - vacapinta, anathema - yes you got the right book (Deep Time by Gregory Benford), and yes, I skipped the final chapter, but the first 2/3 of the book were nonfiction, and really quite interesting.

posted by kokogiak at 11:39 PM on April 15, 2002

posted by kokogiak at 11:39 PM on April 15, 2002

Textism has cool animations "in egregious Flash" of the

Fibonacci Series and the Golden section.

posted by kirkaracha at 11:55 PM on April 15, 2002

Fibonacci Series and the Golden section.

posted by kirkaracha at 11:55 PM on April 15, 2002

When I was looking at building a home theater, a lot of people were using phi to decide on their room sizes and speaker positioning.

posted by bregdan at 4:43 AM on April 16, 2002

posted by bregdan at 4:43 AM on April 16, 2002

The cross section of an Ammonite is one of the best examples of the Golden section in nature, for me at least. When I was little one of the first fossils I found was an ammonite, the rock split just right and there it was, about 5 inches in diameter, a crystal ammonite cast. My dad pointed out how the spiral diminished and that the ratio was that of the golden section. I'm still collecting fossils 2 decades after I first saw it.

posted by jackspot at 7:01 AM on April 16, 2002

posted by jackspot at 7:01 AM on April 16, 2002

joemaller, I wasn't looking for anything specific, I just figured if I mentioned the golden section MeFites would have interesting things to say about it.

posted by lbergstr at 8:04 AM on April 16, 2002

posted by lbergstr at 8:04 AM on April 16, 2002

Yay! More math to explore with POV-Ray! Thanks for the excellent links. I've recently been playing with Buckminster Fuller's geodesic dome geometry and the intrinsically associated cuboctahedron. Really fascinating.

posted by johnnyace at 9:40 AM on April 16, 2002

posted by johnnyace at 9:40 AM on April 16, 2002

The proportions of my undergraduate architectural thesis project (A Museum of Time) are (or, were -- 16 years ago) controlled by the Golden Section. It got so I could identify a Golden Rectangle at one hundred paces (a credit card is actually very close).

I recently built a piece of furniture with proportions involving the Golden Section. [self link]

posted by Dick Paris at 11:19 AM on April 16, 2002

I recently built a piece of furniture with proportions involving the Golden Section. [self link]

posted by Dick Paris at 11:19 AM on April 16, 2002

I think the term you are grasping for when you say "intrinsically associated" is "stellation of", as in, the only stellated form of the octahedron is the stella octangula, which is a compound of two tetrahedra. Maybe not.

Stellation:

The process of constructing polyhedra by extending the facial planes past the edges of a given polyhedron until they intersect.

One fascinating website, by the way, is the everything2 equivilant for mathematics, Mathworld, which is associated with Wolfram Systems and such.

Also, vacapinta, please cast some light on your statement that liquor is "quicker". How in the hell can something be "quicker" than amphetamines? How can you go faster than

posted by Settle at 11:26 AM on April 16, 2002

Stellation:

The process of constructing polyhedra by extending the facial planes past the edges of a given polyhedron until they intersect.

One fascinating website, by the way, is the everything2 equivilant for mathematics, Mathworld, which is associated with Wolfram Systems and such.

Also, vacapinta, please cast some light on your statement that liquor is "quicker". How in the hell can something be "quicker" than amphetamines? How can you go faster than

*speed incarnate*??posted by Settle at 11:26 AM on April 16, 2002

*How can you go faster than speed incarnate??*

Actually, crack cocaine is quicker. Liquor is only quicker than candy.

I've been playing with Robert Abbott's multi-state mazes these days. I mention this because he has created several which involve the cuboctahedron (Fuller's dymaxion) It reminds me when I used to build unfolded hypercubes. I was inspired by Dali's Corpus Hypercubus.

posted by vacapinta at 12:01 PM on April 16, 2002

I'd like to offer a more skeptical view of the Golden Section as put forth in a recent article by mathematician Hans Walser:

posted by krebby at 1:40 PM on April 16, 2002

The belief that the Golden Section has, through the ages, been purposely built into buildings, paintings, or sonatas, or that it is somehow part of the world or of our brains is one that has sprung up fairly recently.The Golden Section is still a number of some mathematical significance, I just don't accept that there's any psychological or aesthetic "magic" in it.

The idea that a rectangle with dimensions Phi (aka Golden Section) and 1 (or, equivalently, 1 and Phi - 1) is the one that is aesthetically most pleasing seems to have gotten a start in the 1860s, though too many authors repeat this as if it were part of the wisdom of the ages. Back then, one Gustav Fechner presented subjects with ten rectangles and asked them which they thought was the nicest. The rectangles varied from a square to one whose sides had the ratio of 2 to 5 -- that is, with aspect ratios from 1 to .4. The three rectangles in the middle, those with aspect ratios .57, .62, and .67, were chosen by 76% of the subjects.

Well, of course. Squares are dull, long flat rectangles look as if they had been stepped on, and tall skinny rectangles make us nervous-they look as if they may fall over any minute-so naturally something in the middle gets picked. But the golden section has nothing to do with it. Further studies have shown that golden rectangles are in fact not the prettiest. There is now (at least there was when this was written) a Web poll on the subject, which may be found at http://homepage.esoterica.pt/~madureir, that has rectangles with width 54 and heights 65, 68, 72, 77, 81, 87, 96, 108. The sixth is close to a golden rectangle, but it has been picked as the most pleasing rectangle by only 12% of the 1501 respondents to the poll. The 54-by-72 rectangle is the clear winner, with 30% of the votes. (The percentages of respondents choosing each of the eight rectangles are, respectively, 9, 3, 30, 16, 19, 12, 5, 8.) If you measure books, a handy source of rectangles, you will find that almost none has dimensions that come close to those of a golden rectangle. A golden-rectangle book looks too tall and skinny.

posted by krebby at 1:40 PM on April 16, 2002

Books are manufactured to non aesthetic and certainly nonbiological constraints, such as readability, ability to divide a standard ream sized paper into such a size, and ability of paper to lay flat.

The golden mean is not holy, certainly. Many people in the past have built it into things believing that it is holy. But one must remember that its significance is not in the fact that it occurs in nature, but what it actually is in itself. The spiral formed by the golden rectangle is one of the only, if not the only spiral which is scaleless. And the golden rectange has a very useful property in how if a square is constructed with one of the shorter segments, the other part is geometrically similar to the original. This is useful stuff to know when you paint things, or design things. It has a property which is totally unique and not insignificant. I find that it provides the best format for a composition that melts away - vertically it produces unimposing portraits and horizontally it produces less grand, more humble landscapes.

Furthermore, you indicate a flaw in the rectangular research, namely that people were asked to choose which rectangle was the most pleasing. Rectangles have character you see, and this is what they responded to. The research is weakened with the assertion that a golden rectangle is too tall and skinny. What if it is on its side? And doesn't the area of the rectange matter? The area of the rectangles people were to choose from was variable. I wouldn't be surprised if they simply chose the size

Anyway. The use of the golden rectangle will often be misguided if you don't know what you're doing. If you keep it in mind, however, you'll find it is one of the safest starting points for many visual solutions.

posted by Settle at 1:55 PM on April 16, 2002

The golden mean is not holy, certainly. Many people in the past have built it into things believing that it is holy. But one must remember that its significance is not in the fact that it occurs in nature, but what it actually is in itself. The spiral formed by the golden rectangle is one of the only, if not the only spiral which is scaleless. And the golden rectange has a very useful property in how if a square is constructed with one of the shorter segments, the other part is geometrically similar to the original. This is useful stuff to know when you paint things, or design things. It has a property which is totally unique and not insignificant. I find that it provides the best format for a composition that melts away - vertically it produces unimposing portraits and horizontally it produces less grand, more humble landscapes.

Furthermore, you indicate a flaw in the rectangular research, namely that people were asked to choose which rectangle was the most pleasing. Rectangles have character you see, and this is what they responded to. The research is weakened with the assertion that a golden rectangle is too tall and skinny. What if it is on its side? And doesn't the area of the rectange matter? The area of the rectangles people were to choose from was variable. I wouldn't be surprised if they simply chose the size

*least*similar to a banner ad.Anyway. The use of the golden rectangle will often be misguided if you don't know what you're doing. If you keep it in mind, however, you'll find it is one of the safest starting points for many visual solutions.

posted by Settle at 1:55 PM on April 16, 2002

Don't click the link in krebby's excerpt - the people are all arguing about the Middle East over there. The poll results to which he refers, though, can be found here.

posted by yhbc at 2:06 PM on April 16, 2002

posted by yhbc at 2:06 PM on April 16, 2002

hey just saw this on boing boing, roger penrose is suing kimberly-clark (kleenex) for using penrose tiling on its toilet paper! btw, tony smith's www home page is pure wackness :)

also, vacapinta you might be interested in beyond the third dimension by thomas banchoff. Settle could take the class! like it's being offered next spring :)

posted by kliuless at 2:11 PM on April 16, 2002

also, vacapinta you might be interested in beyond the third dimension by thomas banchoff. Settle could take the class! like it's being offered next spring :)

posted by kliuless at 2:11 PM on April 16, 2002

"Throughout the semester, you will be given opportunities to use

Sounds pretty interesting. That's exceptionally kind of you to point out kliuless, thanks.

posted by Settle at 2:42 PM on April 16, 2002

*electronic communication tools*(such as email and NetScape)"..Italics theirs.Sounds pretty interesting. That's exceptionally kind of you to point out kliuless, thanks.

posted by Settle at 2:42 PM on April 16, 2002

thanks, kliuless.

Actually, I've spent the last few years on the trail of a madman. Charles H. Hinton, the son-in-law of the mathematician Boole (Boolean logic) built some cubes with which, through a series of exercises, one was supposed to be able to visualize higher dimensions.

Martin Gardner wrote an essay about this called 'Hypercubes', reprinted in his book Mathematical Carnival. He includes a letter from a former user of the cubes warns everyone to stay away from them because they are "completely mind-destroying!". No more information is given.

My goal is to find those cubes.

posted by vacapinta at 3:11 PM on April 16, 2002

Actually, I've spent the last few years on the trail of a madman. Charles H. Hinton, the son-in-law of the mathematician Boole (Boolean logic) built some cubes with which, through a series of exercises, one was supposed to be able to visualize higher dimensions.

Martin Gardner wrote an essay about this called 'Hypercubes', reprinted in his book Mathematical Carnival. He includes a letter from a former user of the cubes warns everyone to stay away from them because they are "completely mind-destroying!". No more information is given.

My goal is to find those cubes.

posted by vacapinta at 3:11 PM on April 16, 2002

Careful vacapinta - sounds like you're developing a taste for

posted by kokogiak at 4:45 PM on April 16, 2002

*Extreme Mathematics*, dude.posted by kokogiak at 4:45 PM on April 16, 2002

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posted by machaus at 5:29 PM on April 15, 2002